Risk and Ambiguity in Models of Business Cycles by David Backus, Axelle Ferriere and Stanley Zin

Discussion Risk and Ambiguity in Models of Business Cycles by David Backus, Axelle Ferriere and Stanley Zin 1 Introduction This is a very interesting, topical and useful paper. The motivation for this paper is the Great recession and its aftermath. An important question that models of business cycles aim to address is what was the major factor that lead to such large output, consumption, investment and employment drops. In this context, uncertainty has been widely reported as a possible cause. The objective of the paper is to investigate how does uncertainty affect the dynamics of macro aggregates in a standard business cycle model. The key ingredients of this model are recursive preferences, ambiguity aversion and technology shocks. Importantly, in this paper, following a recent literature, the concept of uncertainty encompasses not only risk, a situation in which agents know the true probabilities of outcomes, but also ambiguity, or Knightian uncertainty, where agents are not confident in probability assessments. The main finding of this paper is that within the considered business cycle model, the effects of uncertainty, either through risk or ambiguity, are quantitatively small. An important contribution of the paper is to also provide loglinear approximations that in many instances can be used to analytically study the role of preference parameters, such as the intertemporal elasticity of substitution (IES) and uncertainty aversion, for the internal propagation of the model. My discussion consists of several parts. First, I will build a simple two period model to illustrate some of the key properties of the model. Second, I will introduce both risk and ambiguity into that simple two period model and discuss similarities and differences between these two concepts of uncertainty. Third, I will comment on the general ability of uncertainty shocks to be important drivers of business cycles where I focus mostly on their ability to generate positive comovement between the macro aggregates. 2 A simple two period model The notation in this two-period model is that variables with tildes refer to the future (date 2). The date 2 output Ỹ is produced by labor N, chosen in advance at date 1, according to: Ỹ = ZN (1) 1

where Z is a random productivity shock. The market clearing condition states that consumption equals output: C = Ỹ (2) There is a representative agent who derives felicity from consumption (increasing) and hours (decreasing). The agent s problem is a standard intertemporal choice under uncertainty, of finding the optimal N by trading off current disutility of hours against the future utility of consumption. 2.1 Intratemporal and intertemporal uncertainty In discussing the effects of uncertainty on decisions, it is helpful to consider the choice problem in two temporally distinct steps. 2.1.1 Intratemporal variation In the first temporal step, the agent computes the certainty equivalent of the uncertain, from the perspective of date 1, future consumption C. This certainty equivalent, denoted below by µ( C), depends on both sources of intratemporal uncertainty, namely risk and ambiguity. A useful decision theoretical approach to pose the problem of uncertainty is to treat it as a two-stage lottery. 1 In the first stage, nature draws the model, i.e. the probability distribution over outcomes, and in the second stage, conditional on the model, nature draws the outcome. The standard notion of risk applies to the second stage, and aversion to this two-step lottery is usually referred to ambiguity aversion. The special case of ambiguity neutrality would correspond in this setup to reducing the two-stage lottery into one compound lottery, an approach that characterizes models of subjective expected utility used in standard macroeconomic models. Different axiomatizations of ambiguity aversion lead to different ways to consider attitudes and confidence in the probabilities associated to the first-stage lottery. The authors also introduce ambiguity by relating it to two sources of uncertainty and do so through a specific preference representation, namely the smooth ambiguity aversion model of Klibanoff et al. (2005) and its dynamic version of Klibanoff et al. (2009). This axiomatization specifies a unique prior over the probability distributions that constitute the first stage lottery. Aversion to that uncertainty is captured by a concave function of 1 Hansen (2014) uses a similar approach to describe the concerns for model misspecification for agents outside and inside of economic models. 2

the expected utility formed, conditional on a probability distribution, for the second stage lottery. Risk The paper first introduces risk. The attitude towards risk is summarized by the concavity of the function u( C). Conditional on a probability distribution q over Z, the certainty equivalent of this risky lottery, denoted by µ q, is defined as u[µ q ( C)] = E q [u( C)] (3) An example with power utility, where α is the coefficient of relative risk aversion (CRRA) equal to Cu (C)/u (C), leads to the specification µ q ( C) [ ( 1/(1 α) = E C1 α)] q. Ambiguity Consider now uncertainty over the distribution q. Take the certainty equivalent µ q in (3), which conditions on a particular q, and form another certainty equivalent using the probability distribution p, which is now over the distributions q. For this step the authors introduce a new function, v, whose curvature dictates how much the agent dislikes the uncertainty over probability distributions. The overall certainty equivalent resulting from the two types of uncertainty is µ( C) = µ p (µ q ( C)), which satisfies v[µ p (µ q ( C))] = E p [v(µ q ( C))] where an example with power utility, sets µ( C) = [E p ( µ q ( C) 1 γ )] 1/(1 γ). Building on the main application from the paper, where the focus is on ambiguity about conditional means, we can write in this two-period model the two layers of uncertainty as: log Z N(µ 1 2 σ2 z, σ 2 z) (4) which describes the distribution of log Z, conditional on a mean µ. The latter also has a distribution, nromalized to ensure that E( Z) = 1: µ N( 1 2 σ2 µ, σ 2 µ) where the values of the standard deviations σ z and σ µ are known in advance at date 1. We can investigate the effects of uncertainty shocks by analyzing the response of decision rules to changes in σ z or σ µ. 3

2.1.2 Intertemporal variation Having computed the certainty equivalent of the date 2 consumption, in the second temporal step the agent at date 1 intertemporally evaluates the certain consumption stream (C, N, µ( C)). Note that in this example labor at date 2 is optimally constant at zero because of the two period assumption. For more general dynamic problems we would need to solve for continuation utilities which is what makes the solution method an important part of constructing and evaluating dynamic models under uncertainty. In this two period model the preference over intertemporal plans can be described as V (C) θn + βv [µ( C)] (5) where the aversion to intertemporal variation is a function of the curvature in the function V. An example with power utility, where now 1/ρ is the intertemporal elasticity of substitution, defined as V (C)/CV (C), is given by V (C) = (1 ρ) 1 C 1 ρ. 2.2 Decision rule under uncertainty The simplicity of this model can be made to use by employing the specific functional forms. In addition, for analytical convenience we further set θ = β in (5). The date 1 optimal labor then solves the following social planner problem: [ max βn + β(1 N ρ) 1 µ( ZN) ] 1 ρ where we have made use of the market clearing condition (2) and the production function in (1). It follows that the optimal hours trades off the normalized current marginal disutility of labor with its expected marginal benefit. The benefit is evaluated from the perspective of the gain in certainty equivalent terms. The intratemporal uncertainty affects the labor decision through the determination of µ( ZN). Here both risk and ambiguity matter, through the parameters α and γ. The intertemporal effect works through the IES and connects the tradeoff between the certain disutility of today and the certainty equivalent gain of tomorrow. In the special case of γ = α = ρ, we obtain the standard expected utility with time separable power utility in which CRRA is the inverse of the IES and where µ( C) = [E p ( E q C1 α)] 1/(1 α). If furthermore γ = 0, then the uncertainty over the conditional mean does not matter for the certainty equivalent, i.e. the E p term disappears and E q is taken under the average µ = 0. 4

Taking logs and using the log-normality of shocks we obtain the optimal solution log N = 0.5 (1/ρ 1) ( ασ 2 z + γσ 2 µ ) (6) which together with the production function log Ỹ = log Z + log N describe the solution of the model. From the perspective of understanding the effects of ambiguity, the interpretation of (6) is that injecting ambiguity adds to the amount of uncertainty that the agent faces. In fact, a notion of observational equivalence applies. This solution is equivalent to adding risk to the standard case of expected utility. Indeed, the solution with ambiguity is the same as in an economy where the effective risk aversion equals ασ 2 + γχ 2. In the particular case of σ µ = σ z, the equivalent economy is one where there is only risk, with a higher risk aversion coefficient α than in the original economy, i.e. α = α + γ. In other words, in this setup, the agents with smooth ambiguity aversion, facing uncertainty over conditional means, confront ambiguity by acting as if they were more risk averse. 2.3 Links to paper s messages The simple model described above produces the optimal solution for a forward looking decision, such as labor in this case, taken under risk and ambiguity. Formula (6) visualizes some of the messages emphasized in the paper. First, the qualitative business cycle effects of uncertainty depend on the IES. Consider the case of an increase in risk, in the form of a higher conditional variance σz. 2 If uncertainty rises, there are two competing effects on the forward looking labor decision: on the one hand, according to the wealth effect, the agent feels poorer and wants to work more to smooth consumption. On the other hand, the uncertainty-adjusted return of working decreases, so due to this substitution effect, the agent wants to work less. If the IES is large, i.e. if ρ 1 > 1, then the substitution effect is strong enough so that an uncertainty increase generates a recession, in the form of the agent deciding to work less. A similar mechanism works if instead the variance σµ 2 of the conditional mean rises. Second, it shows that the risk aversion coefficient α and ambiguity parameter γ matter quantitatively. They do so by lowering the certainty equivalent. As α or γ increases, it is as if the agent is more pessimistic about the future. This happens naturally here as the agent values less the uncertain future marginal benefit of working more today. Third, in light of the previous points, adding recursive preferences to a business cycle model can help generate larger recessionary consequences of aggregate uncertainty shocks. Indeed, with recursive preferences we can increase risk aversion, which amplifies the quanti- 5

tative effects, while maintaining the IES high enough so produce recessionary effects. In the corresponding model with time separable power utility, it is not possible to simultaneously move risk aversion and IES. Fourth, formula (6) shows that the uncertainty effects depend on the variances σ 2 z and σ 2 µ. This is the usual sense in which, quantitatively, it is likely to find a small impact of uncertainty on decisions. In other words, the common Arrow (1971) local risk neutrality phenomenon, in which the effect of uncertainty on welfare is only second-order, applies in this simple model, as it does in more complicated business cycle models. The fact that locally risk does not matter for decisions, or alternatively that its effect enters through second order terms such as the variance, is independent of the solution method. It is obvious in a linearized version of the optimality condition, where the effect is zero, and at the same is clear also in the log-linearized version as in (6). For the present model of ambiguity aversion, the same local result also holds. Fifth, ambiguity can magnify effects of standard risk shocks and/or can be an independent source of fluctuations. The paper provides several examples of time-variations in uncertainty: one is called Model 1, where realized volatility also directly affects the amount of uncertainty over the conditional mean. 2 In the notation of the two-period model, this amounts to setting σ µ,t = σ z,t. A second setup is to consider learning. In the paper such an example is provided where the observed technology shock is used to infer the unobserved conditional mean. A feature of that setup is that the posterior variance, and thus ambiguity, will deterministically converge to a constant. In an alternative Model 2, only ambiguity is stochastic. In the above notation this corresponds to making σ 2 µ,t stochastic while keeping σ z constant. A sixth point concerns the solution method. The paper follows a tradition in the finance literature of studying variation in uncertainty with log-linear decision rules, while also quantifying the model s properties with a more accurate global solution. A clear advantage is the analytical tractability and the transparency that comes with it. The simple model presented above is exactly log-linear, but in general this approach requires an approximation. The main functional form assumptions that are needed here are the log-normality of shocks and the linearity of volatility. A final observation is that in a business cycle model with risk and ambiguity we can meaningfully talk about asset prices and the corresponding uncertainty premia. In the simple model derived here, one can find the price of a one-step-ahead consumption claim to be 2 Bidder and Smith (2012) and Bianchi et al. (2014) are other recent examples that link stochastic volatility to the amount of ambiguity, through the multiplier preferences of Hansen and Sargent (2008) and the recursive multiple priors utility of Epstein and Schneider (2003), respectively. 6

( ) R f 1 E ( C)e ασ2 z γσµ 2 where R f denotes the risk free rate and E ( C) denotes expectation under the average µ = 0. Although asset prices are less of a focus in this paper, the authors do mention basic properties of the pricing kernel. 3 The key observation here is that ambiguity offers a theoretically relevant concept that may help explain observed movements in compensations for uncertainty, for example from asset market data, without having to appeal only to second moment changes. Indeed, an econometrician would recover excess returns over the risk free rate that can come from either risk or ambiguity. 4 2.4 An alternative model of ambiguity aversion The decision theory literature has proposed alternative representations of ambiguity aversion. One that has also been recently used in asset pricing and business cycle models follows the multiple priors model of Gilboa and Schmeidler (1989) and its dynamic version of Epstein and Schneider (2003). Here the agent deals with this uncertainty by considering a set P of beliefs over the future TFP. The size of the set is a primitive that characterizes the agent s lack of confidence. The certainty equivalent in this case is obtained by evaluating the expected value of u( C) as if the worst-case probability distribution characterizes the true data generating process: u[µ( C)] = min P P EP [u( C)] (7) Thus, here the lack of confidence in the first-stage lottery is reflected in the min operator, whereas the risk associated with the second-stage lottery, conditional on a distribution P, is still summarized by the concavity of the u( C) function. If we focus attention on conditional means, then log Z is still described as in (4), but the agent is not confident enough to assign a unique distribution to the mean µ, which instead is allowed to belong to a specified set µ [ a, a]. The true data generating process in this case is assumed to be described by setting µ = 0 in (4). Given that utility is increasing in consumption and the latter is in equilibrium simply equal to ZN, then the solution to the minimization problem in (7) is to use the worst-case conditional mean γ = a. The interpretation here is that the uncertainty associated with the probability distributions makes the social planner act as if future technology is lower. Using the same functional forms as in the case of risk, the optimal labor choice is log N = (1/ρ 1) ( 0.5ασ 2 z + a ) 3 A more in depth analysis of the asset pricing side is made in the authors previous work, such as Backus et al. (2014). 4 Bianchi et al. (2014) builds and estimates a business cycle model to jointly explain time-variation in macro aggregates, excess returns and asset supply using regimes switches in both volatility and ambiguity. 7

which replaces the smooth ambiguity effect in (6) by the worst-case conditional mean. In other words, the agent with multiple priors utility, facing uncertainty over conditional means, confronts ambiguity by acting as if the conditional mean is lower. A feature of this preference is that, due to the lower conditional mean argument, the welfare effects of ambiguity can be first-order. 5 3 Uncertainty shocks and business cycles 3.1 Positive comovement of macro aggregates One of the most salient cyclical property that characterizes business cycles is the positive comovement across macro aggregates, such as consumption, hours and investment. Thus, for a shock to have the potential to be an important driver of business cycles it seems crucial to be able to generate such comovement. In evaluating whether uncertainty shocks fulfill this criterion, the authors correctly point to the Barro and King (1984) critique. It is useful in this context, to briefly recall what that critique is. First, it should be noted that in the business cycle model studied in this paper, labor supply is constant. So there is really no intratemporal margin to change production. Output is given by Y t = C t + I t = A t K ω t 1 where C t, I t, K t and A t denote consumption, investment, capital and TFP, respectively. By construction, in response to any other shock than TFP, such as an uncertainty shock, C t and I t comove negatively. To give these other shocks a chance to produce comovement we need to introduce some intratemporal margin. The obvious choice would be flexible labor. Let us start by considering a standard model with time-additive utility U(C t, 1 L t ), where L t is hours worked. The equilibrium is augmented by a standard intratemporal optimality characterizing optimal labor: χ t U 2 (C t, 1 L t ) U 1 (C t, 1 L t ) = F L(A t, K t 1, L t ) (8) where the marginal rate of substitution (MRS), possibly affected by a disutility of labor shock χ t, equals the marginal product of labor. 5 A comparison between various types of ambiguity aversion has been in explored in some detail in the literature (see Epstein and Schneider (2010) and Guidolin and Rinaldi (2013) for examples of overviews), covering mostly decision-theory perspectives and applications to the finance literature. 8

In this context, we can investigate whether C t and L t can comove positively with uncertainty shocks. The Barro-King logic gives a clear negative answer. The argument is the following. Holding constant the marginal product of labor, the MRS in (8) increases in both C t and L t. Thus, we can only get both to go up if the marginal product increases. But without TFP shocks, the latter actually declines. According to this argument, we need intratemporal shocks to generate comovement. They can be either in the form of TFP shocks, which would affect the intratemporal choice through the marginal product of labor, or MRS shocks χ t, broadly referred to in the literature as labor supply shocks. The Barro-King critique is usually stated for time-additive preferences. It is then interesting to analyze whether recursive preferences, as used in this paper, are able to overturn the comovement problem. The short answer is again negative. To see this, take a functional form for the per period utility to be a composite of c t and 1 L t of the form V t = [(1 β)u ρ t + βµ t (U t+1 ) ρ ] 1/ρ ; u t = c υ t (1 L t ) 1 υ and let output be Y t = A t K ω t 1L 1 ω t. The key observation is that the labor intratemporal optimality is really the same as with additive utility 1 υ υ c t = (1 ω) A t K 1 L t 1L ω ω t t which makes the Barro-King logic apply again. The reason is that the recursive terms, while contributing to intertemporal conditions, do not affect the MRS between c t and L t. So, within the context of standard business cycle models, even with recursive preferences, uncertainty shocks cannot generate positive comovement. However, there are model elements that can overturn this conclusion. For example, adding features that affect the MRS, such as internal habit formation or GHH preferences with low wealth effects, together with investment adjustment costs are likely to generate comovement out of uncertainty shocks 6. Another venue is to add nominal rigidities. For example, with sticky prices, the model produces endogenous countercyclical markup, that shows up as an extra term that inversely affects the marginal product of labor in equation (8). This rigidity can thus act as a force that breaks the Barro-King impossibility result and it has been applied recently for volatility and ambiguity shocks, for example by Basu and Bundick (2011) and Ilut and Schneider (2014). 6 See Christiano et al. (2010) or Jaimovich and Rebelo (2009) for example of such models. 9

3.2 Ambiguity shocks driving business cycles This discussion has pointed to some modeling choices, such as the representation of ambiguity aversion and model features that can generate positive comovement out of uncertainty shocks, that are likely to affect the conclusion whether uncertainty shocks can be an important driver of business cycle. There are two further points that are worth noting here. One is the solution method. Axiomatizations that work through higher order effects usually require higher order approximations, which in turn make estimation difficult. 7 This current paper is closer to the finance literature in that it employs a log-linear approximation within the context of affine models. Malkhozov (2014) analyzes the connection between this approach and standard perturbation techniques and shows that the effects of volatility shocks are similar. In particular, the former produces the same effect as the third-order perturbation. There are preference representations, such as the recursive multiple priors, that allow for linear, first-order, effects of uncertainty shocks. In this case, a standard linear approximation can be easily applied. An advantage of linearized models is that is allows the model to be estimated using standard estimation techniques, such as a Kalman filter, likelihood- based approach, that have been used extensively in quantitative macroeconomics analysis. The second further point is related to the magnitude of the ambiguity shocks. The paper injects a type of ambiguity in the business cycle model that relies on two parameters: one is the aversion to ambiguity, captured by γ, and another use is the amount of variance over the conditional mean, denoted by σ 2 µ in this discussion. It would be useful to assess whether this leads to large or small ambiguity, using some metric. For example Hansen and Sargent (2008) use detection error probabilities to discipline the size of the worst-case probability distribution. In a similar spirit, but using non-stationarity arguments, Ilut and Schneider (2014) relate the set of beliefs to the statistical property of the forecasts based on the worst-case belief. The authors are careful to state that they do not find large effects of uncertainty, either through risk or ambiguity, conditional on the standard business cycle model that they study. This discussion has highlighted some alternative model ingredients that may make these effects much more significant. As an example of such an approach, Ilut and Schneider (2014) study and estimate a simpler version of the Christiano et al. (2005) model, and find that: (i) qualitatively, ambiguity shocks can act as labor supply shocks, and thus generate positive comovement, and (ii) when using dispersion of SPF forecasts on real GDP growth to discipline the process for the time-varying ambiguity, ambiguity shocks can account for most of the fluctuations in US aggregate consumption, hours worked and investment. 7 Solution methods based on higher-order approximations typically require non-linear filters, such as the particle filter. See Fernández-Villaverde et al. (2010) for a review of methods in macro-econometrics. 10

4 Conclusion Uncertainty is a major factor in forward-looking decisions. This paper takes seriously the notion that uncertainty is not just risk but also ambiguity and studies both concepts in a standard business cycle model. The main finding is that, within the context of these models, uncertainty is not likely to be an important factor in business cycle fluctuations. The present discussion has focused on the qualitative mechanisms through which uncertainty affects decision rules and suggested that modeling choices are likely to affect significantly the conclusion on the role of uncertainty. This paper is part of a growing literature on uncertainty and business cycles. There are many further issues of interest in this area, some of which include: linking macro aggregates and asset prices, estimating models of stochastic volatility and ambiguity, accounting for different frequencies of uncertainty or modeling endogenous variations in uncertainty. References Arrow, K. J. (1971): The theory of risk aversion, Essays in the theory of risk-bearing, 90 120. Backus, D., M. Chernov, and S. Zin (2014): Sources of entropy in representative agent models, The Journal of Finance, 69, 51 99. Barro, R. J. and R. G. King (1984): Time-separable preferences and intertemporalsubstitution models of business cycles, Quarterly Journal of Economics, 99. Basu, S. and B. Bundick (2011): Uncertainty Shocks in a Model of Effective Demand, Boston College Working Papers 774. Bianchi, F., C. Ilut, and M. Schneider (2014): Uncertainty shocks, asset supply and pricing over the business cycle, NBER Working Papers 20081. Bidder, R. and M. E. Smith (2012): Robust Animal Spirits, Journal of Monetary Economics, 59, 738 750. Christiano, L. J., M. Eichenbaum, and C. L. Evans (2005): Nominal Rigidities and the Dynamic Effects of a Shock to Monetary Policy, Journal of Political Economy, 113. Christiano, L. J., C. L. Ilut, R. Motto, and M. Rostagno (2010): Monetary Policy and Stock Market Booms, NBER Working Papers 16402. 11

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